Group #37574127 — subgroups

Subgroups — C₂×(C₂×D₁₉₁)⋊C₁₆ #37574127

Order 24448.

272 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₉₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₁₉₁)⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×D₁₉₁	C₄	1	1	yes	characteristic
6	C₂²×C₁₉₁⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₉₁	—	2	2	no	—
8	C₂×C₈×D₁₉₁	—	2	2	no	—
9	C₂×C₈×D₁₉₁	C₄	1	2	yes	—
10	C₂×C₈×D₁₉₁	C₂²	1	2	yes	—
11	C₂×C₁₉₁⋊C₁₆	—	2	2	no	—
12	C₂×C₁₉₁⋊C₁₆	C₂²	1	2	yes	complemented
13	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₉₁	—	2	2	no	—
15	C₂×C₁₆×C₁₉₁	C₂²	1	2	yes	complemented
16	C₂³×D₁₉₁	C₈	1	1	yes	characteristic
17	C₂²×Dic₁₉₁	C₈	1	1	yes	characteristic
18	C₂×C₄×D₁₉₁	—	2	2	no	—
19	C₂×C₄×D₁₉₁	—	2	2	no	—
20	C₂×C₄×D₁₉₁	C₈	1	2	yes	—
21	C₂×C₄×D₁₉₁	C₂×C₄	1	2	yes	—
22	C₂×C₁₉₁⋊C₈	—	2	2	no	—
23	C₂×C₁₉₁⋊C₈	C₂×C₄	1	2	yes	—
24	C₈×D₁₉₁	—	2	4	no	—
25	C₈×D₁₉₁	—	2	2	no	—
26	C₈×D₁₉₁	—	2	2	no	—
27	C₁₉₁⋊C₁₆	—	2	2	no	—
28	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
31	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
32	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
33	C₁₆×C₁₉₁	—	2	2	no	—
34	C₂²×D₁₉₁	—	2	2	no	—
35	C₂²×D₁₉₁	—	2	2	no	—
36	C₂²×D₁₉₁	C₁₆	1	4	yes	complemented
37	C₂²×D₁₉₁	C₂×C₈	1	2	yes	—
38	C₂×Dic₁₉₁	—	2	2	no	—
39	C₂×Dic₁₉₁	C₂×C₈	1	2	yes	—
40	C₄×D₁₉₁	—	2	4	no	—
41	C₄×D₁₉₁	—	2	2	no	—
42	C₄×D₁₉₁	—	2	2	no	—
43	C₁₉₁⋊C₈	—	2	2	no	—
44	C₂³×C₁₉₁	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₉₁	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₁₉₁	M₁₆	1	2	yes	—
48	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
49	C₈×C₁₉₁	C₂²⋊C₄	1	2	yes	—
50	C₈×C₁₉₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₉₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₁₉₁	C₂²⋊C₈	1	2	yes	—
53	C₂²×C₁₉₁	C₂×C₁₆	1	2	yes	—
54	C₂²×C₁₉₁	M₃₂	1	2	yes	—
55	C₂²×C₁₉₁	C₂²×C₈	1	1	yes	characteristic
56	C₂×D₁₉₁	—	2	8	no	—
57	C₂×D₁₉₁	—	2	2	no	—
58	C₂×D₁₉₁	—	2	2	no	—
59	C₂×D₁₉₁	C₂×C₁₆	1	4	yes	complemented
60	C₄×C₁₉₁	C₂²⋊C₈	1	2	yes	—
61	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₁₉₁	C₂×M₁₆	1	1	yes	characteristic
63	Dic₁₉₁	—	2	2	no	—
64	C₂×C₁₉₁	C₂²⋊C₁₆	1	4	yes	complemented
65	C₂×C₁₉₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
66	C₂×C₁₉₁	C₂²×C₁₆	1	1	yes	characteristic
67	C₂×C₁₉₁	C₂×M₃₂	1	1	yes	characteristic
68	D₁₉₁	—	2	4	no	—
69	C₁₉₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
70	C₂×C₂²⋊C₁₆	—	191	1	no	maximal, Sylow
71	C₂³×C₈	—	191	1	no	—
72	C₂²×C₁₆	—	191	1	no	—
73	C₂²×C₁₆	D₁₉₁	1	1	yes	characteristic, complemented
74	C₂²⋊C₁₆	—	191	4	no	—
75	C₂³×C₄	—	191	1	no	—
76	C₂²×C₈	—	382	2	no	—
77	C₂²×C₈	—	382	2	no	—
78	C₂²×C₈	—	191	1	no	—
79	C₂²×C₈	—	191	2	no	—
80	C₂²×C₈	—	191	2	no	—
81	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	—	382	2	no	—
84	C₂×C₁₆	—	191	2	no	—
85	C₂×C₁₆	C₂×D₁₉₁	1	2	yes	complemented
86	C₂⁴	—	191	1	no	—
87	C₂²×C₄	—	382	2	no	—
88	C₂²×C₄	—	382	2	no	—
89	C₂²×C₄	—	191	1	no	—
90	C₂²×C₄	—	191	2	no	—
91	C₂²×C₄	—	191	2	no	—
92	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
93	C₂×C₈	—	382	2	no	—
94	C₂×C₈	—	382	4	no	—
95	C₂×C₈	—	382	2	no	—
96	C₂×C₈	—	382	2	no	—
97	C₂×C₈	—	191	2	no	—
98	C₂×C₈	C₄×D₁₉₁	1	1	yes	characteristic
99	C₂×C₈	C₂²⋊D₁₉₁	1	2	yes	—
100	C₂×C₈	D₇₆₄	1	2	yes	—
101	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
102	C₁₆	—	2	2	no	—
103	C₁₆	—	382	2	no	—
104	C₂³	—	382	2	no	—
105	C₂³	—	382	2	no	—
106	C₂³	—	191	4	no	—
107	C₂³	—	191	2	no	—
108	C₂³	C₈×D₁₉₁	1	1	yes	characteristic
109	C₂×C₄	—	382	2	no	—
110	C₂×C₄	—	382	4	no	—
111	C₂×C₄	—	382	2	no	—
112	C₂×C₄	—	382	2	no	—
113	C₂×C₄	—	191	2	no	—
114	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
115	C₂×C₄	C₈×D₁₉₁	1	1	yes	characteristic
116	C₂×C₄	#15690270	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
118	C₈	—	382	2	no	—
119	C₈	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
120	C₈	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
121	C₈	C₂×D₇₆₄	1	1	yes	characteristic
122	C₂²	—	382	8	no	—
123	C₂²	—	382	2	no	—
124	C₂²	—	382	2	no	—
125	C₂²	—	191	4	no	—
126	C₂²	(C₂×D₁₉₁)⋊C₈	1	2	yes	—
127	C₂²	C₁₆×D₁₉₁	1	2	yes	—
128	C₂²	#20259261	1	2	yes	—
129	C₂²	C₂×C₈×D₁₉₁	1	1	yes	characteristic
130	C₄	—	382	2	no	—
131	C₄	(C₂×D₁₉₁)⋊C₈	1	2	yes	—
132	C₄	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
133	C₄	C₂×#15690270	1	1	yes	characteristic
134	C₂	—	382	4	no	—
135	C₂	(C₂×D₁₉₁)⋊C₁₆	1	4	yes	direct product factor
136	C₂	C₂×(C₂×D₁₉₁)⋊C₈	1	1	yes	characteristic
137	C₂	C₂×C₁₆×D₁₉₁	1	1	yes	characteristic
138	C₂	C₂×#20259261	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₉₁)⋊C₁₆

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₉₁

C₄

yes

characteristic

C₂²×C₁₉₁⋊C₈

C₄

yes

characteristic

C₂×C₈×D₁₉₁

—

C₂×C₈×D₁₉₁

—

C₂×C₈×D₁₉₁

C₄

yes

—

C₂×C₈×D₁₉₁

C₂²

yes

—

C₂×C₁₉₁⋊C₁₆

—

C₂×C₁₉₁⋊C₁₆

C₂²

yes

complemented

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×C₁₆×C₁₉₁

—

C₂×C₁₆×C₁₉₁

C₂²

yes

complemented

C₂³×D₁₉₁

C₈

yes

characteristic

C₂²×Dic₁₉₁

C₈

yes

characteristic

C₂×C₄×D₁₉₁

—

C₂×C₄×D₁₉₁

—

C₂×C₄×D₁₉₁

C₈

yes

—

C₂×C₄×D₁₉₁

C₂×C₄

yes

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

—

C₈×D₁₉₁

—

C₈×D₁₉₁

—

C₈×D₁₉₁

—

C₁₉₁⋊C₁₆

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

D₄

yes

—

C₂×C₈×C₁₉₁

D₄

yes

—

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₁₆×C₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

C₁₆

yes

complemented

C₂²×D₁₉₁

C₂×C₈

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂×C₈

yes

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₁

M₁₆

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₈×C₁₉₁

C₂²⋊C₄

yes

—

C₈×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

C₂×D₄

yes

characteristic

C₂²×C₁₉₁

C₂²⋊C₈

yes

—

C₂²×C₁₉₁

C₂×C₁₆

yes

—

C₂²×C₁₉₁

M₃₂

yes

—

C₂²×C₁₉₁

C₂²×C₈

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

C₂×C₁₆

yes

complemented

C₄×C₁₉₁

C₂²⋊C₈

yes

—

C₄×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₉₁

C₂×M₁₆

yes

characteristic

Dic₁₉₁

—

C₂×C₁₉₁

C₂²⋊C₁₆

yes

complemented

C₂×C₁₉₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₉₁

C₂²×C₁₆

yes

characteristic

C₂×C₁₉₁

C₂×M₃₂

yes

characteristic

D₁₉₁

—

C₁₉₁

C₂×C₂²⋊C₁₆

yes

characteristic, complemented, Sylow

C₂×C₂²⋊C₁₆

—

191

maximal, Sylow

C₂³×C₈

—

191

—

C₂²×C₁₆

—

191

—

C₂²×C₁₆

D₁₉₁

yes

characteristic, complemented

C₂²⋊C₁₆

—

191

—

C₂³×C₄

—

191

—

C₂²×C₈

—

382

—

C₂²×C₈

—

382

—

C₂²×C₈

—

191

—

C₂²×C₈

—

191

—

C₂²×C₈

—

191

—

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

382

—

C₂×C₁₆

—

191

—

C₂×C₁₆

C₂×D₁₉₁

yes

complemented

C₂⁴

—

191

—

C₂²×C₄

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₂×C₈

C₄×D₁₉₁

yes

characteristic

C₂×C₈

C₂²⋊D₁₉₁

yes

—

100

C₂×C₈

D₇₆₄

yes

—

101

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

102

C₁₆

—

103

C₁₆

—

382

—

104

C₂³

—

382

—

105

C₂³

—

382

—

106

C₂³

—

191

—

107

C₂³

—

191

—

108

C₂³

C₈×D₁₉₁

yes

characteristic

109

C₂×C₄

—

382

—

110

C₂×C₄

—

382

—

111

C₂×C₄

—

382

—

112

C₂×C₄

—

382

—

113

C₂×C₄

—

191

—

114

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

115

C₂×C₄

C₈×D₁₉₁

yes

characteristic

116

C₂×C₄

#15690270

yes

—

117

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

118

C₈

—

382

—

119

C₈

(C₂×D₁₉₁)⋊C₄

yes

—

120

C₈

C₂×C₂²⋊D₁₉₁

yes

characteristic

121

C₈

C₂×D₇₆₄

yes

characteristic

122

C₂²

—

382

—

123

C₂²

—

382

—

124

C₂²

—

382

—

125

C₂²

—

191

—

126

C₂²

(C₂×D₁₉₁)⋊C₈

yes

—

127

C₂²

C₁₆×D₁₉₁

yes

—

128

C₂²

#20259261

yes

—

129

C₂²

C₂×C₈×D₁₉₁

yes

characteristic

130

C₄

—

382

—

131

C₄

(C₂×D₁₉₁)⋊C₈

yes

—

132

C₄

C₂×(C₂×D₁₉₁)⋊C₄

yes

characteristic

133

C₄

C₂×#15690270

yes

characteristic

134

C₂

—

382

—

135

C₂

(C₂×D₁₉₁)⋊C₁₆

yes

direct product factor

136

C₂

C₂×(C₂×D₁₉₁)⋊C₈

yes

characteristic

137

C₂

C₂×C₁₆×D₁₉₁

yes

characteristic

138

C₂

C₂×#20259261

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₉₁
commutator	C₂×C₁₉₁	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₁₉₁
Fitting	C₂²×C₁₆×C₁₉₁	C₂
socle	C₂³×C₁₉₁	C₂×C₈
prime radical	C₂³×D₁₉₁	C₈
cosocle	C₂×C₈×C₁₉₁	C₂³
hypercenter	C₂²×C₁₆	D₁₉₁
hypocenter	C₁₉₁	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²×C₈

C₂×D₁₉₁

commutator

C₂×C₁₉₁

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₁₉₁

Fitting

C₂²×C₁₆×C₁₉₁

C₂

socle

C₂³×C₁₉₁

C₂×C₈

prime radical

C₂³×D₁₉₁

C₈

cosocle

C₂×C₈×C₁₉₁

C₂³

hypercenter

C₂²×C₁₆

D₁₉₁

hypocenter

C₁₉₁

C₂×C₂²⋊C₁₆

LCS

G ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. C₂²×C₁₉₁⋊C₁₆	C₂	semidirect
3. (C₂×D₁₉₁)⋊C₁₆	C₂	direct (N × H)
4. C₂²×C₁₆×C₁₉₁	C₂	semidirect
12. C₂×C₁₉₁⋊C₁₆	C₂²	semidirect
15. C₂×C₁₆×C₁₉₁	C₂²	semidirect
36. C₂²×D₁₉₁	C₁₆	semidirect
59. C₂×D₁₉₁	C₂×C₁₆	semidirect
64. C₂×C₁₉₁	C₂²⋊C₁₆	semidirect
69. C₁₉₁	C₂×C₂²⋊C₁₆	semidirect
73. C₂²×C₁₆	D₁₉₁	semidirect
85. C₂×C₁₆	C₂×D₁₉₁	semidirect
135. C₂	(C₂×D₁₉₁)⋊C₁₆	direct (N × H)

H ≅ G/N

type

2. C₂²×C₁₉₁⋊C₁₆

C₂

semidirect

3. (C₂×D₁₉₁)⋊C₁₆

C₂

direct (N × H)

4. C₂²×C₁₆×C₁₉₁

C₂

semidirect

12. C₂×C₁₉₁⋊C₁₆

C₂²

semidirect

15. C₂×C₁₆×C₁₉₁

C₂²

semidirect

36. C₂²×D₁₉₁

C₁₆

semidirect

59. C₂×D₁₉₁

C₂×C₁₆

semidirect

64. C₂×C₁₉₁

C₂²⋊C₁₆

semidirect

69. C₁₉₁

C₂×C₂²⋊C₁₆

semidirect

73. C₂²×C₁₆

D₁₉₁

semidirect

85. C₂×C₁₆

C₂×D₁₉₁

semidirect

135. C₂

(C₂×D₁₉₁)⋊C₁₆

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₈×D₁₉₁	2
C₂²×C₁₉₁⋊C₁₆	2
(C₂×D₁₉₁)⋊C₁₆	2
C₂²×C₁₆×C₁₉₁	2
C₂×C₂²⋊C₁₆	191

subgroup

index

C₂²×C₈×D₁₉₁

C₂²×C₁₉₁⋊C₁₆

(C₂×D₁₉₁)⋊C₁₆

C₂²×C₁₆×C₁₉₁

C₂×C₂²⋊C₁₆

191

Subgroups — C₂×(C₂×D₁₉₁)⋊C₁₆ #37574127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×(C2×D191)⋊C16 #37574127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₂×D₁₉₁)⋊C₁₆ #37574127